Vector Calculus: Understanding Flux

Once you understand flux intuitively, you don’t need to memorize equations. The formulas become “obvious” dare I say. However, it took a lot of effort to truly understand that:

Flux is the amount of “something” (electric field, bananas, whatever you want) passing through a surface.

The total flux depends on strength of the field, the size of the surface it passes through, and their orientation.

Your vector calculus math life will be so much better once you understand flux. And who doesn’t want that?

Physical Intuition

Think of flux as the amount of something crossing a surface. This “something” can be water, wind, electric field, bananas, pretty much anything you can imagine. Math books will use abstract concepts like electric fields, which is pretty hard to visualize. I find bananas more memorable, so we’ll be using those.

To measure the flux (i.e. bananas) passing through a surface, we need to know

The surface you are considering (shape, size and orientation)

The source of the flux (strength of the field, and which way it is spitting out bananas flux)

The strength of the field is important – would you rather have a handful of $5 or $20 bills “flux” into your bank account? Would you rather have a big or little banana come your way? No need to answer that one.

Background Ideas

Keep a few ideas in mind when considering flux:

Vector Field: This is the source of the flux: the thing shooting out bananas, or exerting some force (like gravity or electromagnetism). Flux doesn’t have to be a physical object — you can measure the “pulling force” exerted by a field.

Surface: This is the boundary the flux is crossing through or acting on. The boundary could be a sphere, a plane, even the top of a bucket. Notice that the boundary may not exist — the top of a bucket traces out a circle, but the hole isn’t actually there. We’re considering the flux passing through the region the circle defines.

Timing: We measure flux at a single point in time. Freeze time and ask “Right now, at this moment, how much stuff is passing through my surface?”. If your field doesn’t change over time, then all is well. If your field does change, then you need to pick a point in time to measure the flux.

Measurement: Flux is a total, and is not “per unit area” or “per unit volume”. Flux is the total force you feel, the total number of bananas you see flying by your surface. Think of flux like weight. (There is a separate idea of "flux density" (flux/volume) called divergence, but that’s a separate article).

Flux Factors

The source of flux has a huge impact on the total flux. Doubling the source (doubling the “banana-ness” of each banana), will double the flux passing through a surface.

Total flux also depends on the orientation of the field and the surface. When our surface completely faces the field it captures maximum flux, like a sail facing directly into the wind. As the surface tilts away from the field, the flux decreases as less and less flux crosses the surface.

Eventually, we get zero flux when the source and boundary are parallel — the flux is passing over the boundary, but not crossing through it. It would be like holding a bucket sideways under a waterfall. You wouldn’t capture much water (ignoring splashing) and may get a few funny looks.

Total flux also depends on the size of our surface. In the same field, a bigger bucket will capture more flux than a smaller one. When we figure out our total flux, we need to see how much field is passing through our entire surface.

This is simple stuff so far, right? If you forget, just think about capturing water from a waterfall. What matters? The strength of the waterfall, the size of the bucket and the orientation of the bucket.

Positive and Negative Flux

One last detail – we need to decide on a positive and negative direction for flux. This decision is arbitrary, but by convention (aka your math teacher will penalize you if you don’t agree), positive flux leaves a closed surface, and negative flux enters a closed surface.

Think of flux as a hose spraying water. Positive flux means flux is leaving the hose; the hose is a source of flux. Negative flux is like water entering a sink; it is a sink of flux. So positive flux = leaving, negative = entering. Got it? (By the way, the terms “source” and “sink” are sometimes used to describe fields).

Quick Summary

Quick checkpoint: Flux depends on

The size of the surface

Magnitude of the source field

The angle between them

A fire hose shooting at a tiny bucket (small surface, large magnitude) could have the same flux as a garden hose aimed at a large bucket (large surface, small magnitude). And in case you forgot, flux reminds us to hold the bucket so it is facing the source. This should be obvious – but don’t you want ideas (especially in math!) to be obvious?

Math Intuition

Now that we have a physical intuition, let’s try to derive the math. In most cases, the source of flux will be described as a vector field: Given a point (x,y,z), we’ll get a vector describing the flux.

We want to know how much of that vector field is acting/passing through our surface, taking the magnitude, orientation, and size into account. From our intuition, it should look something like this:

Total flux = Field Strength * Surface Size * Surface Orientation

However, this formula only works if the vector field is the same at every point. Usually, it’s not, so we’ll take the standard calculus approach to solving problems:

Divide the surface into pieces

Find the flux at each piece

Add up the small units of flux to get total flux (integrate).

Let’s go out on a limb and call the tiny piece of the surface dS. Total flux is:

Total flux = (Field Strength * dS * Orientation) for every dS.

or

Total flux = Integral (Field Strength * Orientation * dS)

Make sense so far? Now, we need to figure out how much orientation actually matters. Like we said before, if the field and the surface are parallel, then there is zero flux. If they are perpendicular, there is full flux.

(In this diagram, the flux is parallel with the top surface, and nothing enters from that direction. Mathematically, we represent surfaces by their normal vector, which sticks out of the surface. Don’t let this bookkeeping detail disrupt your visualization.)

If there is an angle, then it is some factor in-between:

How much, exactly? Well, this is a job for the dot product, which is the projection of the field onto the surface. The dot product gives us a number (from 0 to 1) that tells us what percent of the field is passing through the surface. So, the equation becomes:

Total flux = Integral( Vector Field Strength dot dS )

And finally, we convert to the stuffy equation you’ll see in your textbook, where F is our field, S is a unit of area and n is the normal vector of the surface:

Time for one last detail — how do we find the normal vector for our surface?

Good question. For a surface like a plane, the normal vector is the same in every direction. For a sphere, the normal vector is in the same direction as r, the position vector: the top of a sphere has a normal vector that goes out the top; the bottom has one going out the bottom, etc.

More complicated shapes may have a normal vector that varies quite a bit. In this case, try to break the shape into smaller regions (like spheres, cylinders and planes) and find the flux in each part. Then, add up the flux in each region to get the total flux (keeping in mind positve and negative flux).

If the shape is more complicated than that, you may need a computer model or more advanced theorems; but at least you know what is happening behind the scenes.

Flux Examples

Let’s do a few thought experiments to understand flux. Imagine a tube, that lets water pass right through it. We hold the tube under a waterfall, wait a few seconds, then ask what the flux is. I want a numeric answer – what is the flux?

You might think we need to know the speed of the waterfall, the size of the tube, the orientation, etc. But that isn’t the case.

Remember our convention for flux orientation: positive means flux is leaving, negative means flux is entering. In this example, water is falling downward, or entering the tube. This means the top surface has negative flux (it appears to be siphoning up water).

However, what’s happening at the bottom of the box? The water passed through the top and is now leaving the bottom, which is positive flux:

Ah, this beautiful diagram shows what is going on. The top of the box / tube says that water is entering, and the bottom says water is leaving. Assuming the same amount of water is leaving and entering (the rate of water falling is a constant), the net flux would be zero. Think of it as X + (-X) = 0.

What if we had increased the rate of water? Decreased? What would happen?

My (possibly incorrect) answer: If we increased the rate, it means more water would enter than leaves, for a brief moment. We’d have a momentary spike in negative flux (the tube would look like a sink), until the rates equalized. Vice versa if we decreased the rate of water – we’d have a brief spike of positive flux (more water was leaving than entering), until the rate equalized.

Even though net flux is zero, this is different from having zero flux pass through each surface. If you are in an empty field, no shape will generate any flux. But if you are in a field where flux is canceling, changing your shape or orientation could create a non-zero flux. Recognize the difference between having zero flux because the field is zero, vs. having all the flux cancel.

One more point – the “tube” we are considering is a region we define, not a physical tube. Measuring flux is about drawing imaginary boundaries, not having a physical shape. So, when we define the region of a “bucket”, it would not “fill up” with flux. Flux is what is passing through the sides of a bucket at a moment in time. Clearly, if we put in a physical bucket it would fill up, but that’s not what we’re measuring. We’re seeing how much flux would be entering a region we define, from any and all sides (not just the opening). Got it?

And one more point. We haven’t really talked about the units of flux. What is it measured in? As far as I understand, the units can be anything – it depends on the unit of your vector field. So, your vector field might represent bananas, in which case you get total bananas crossing a surface. Or, your field could represent bananas-per-second, in which case you’d get the bananas-per-second crossing your surface. The units of flux depend on the units of your vector field.

Flux is relatively simple to understand, and is really helpful in vector calculus and physics. Trying to understand flux by looking at a mess of integrals is not the way to go. First get an intuitive understanding, and the details will make more sense.

Insights

Here’s a few insights that hit me after learning about flux:

You can take the time derivative of flux. If the vector field (F) changes with time (t), you can use dF/dt to see how the total flux changes over time. Even though flux is taken at a unit in time, you can measure flux at two consecutive moments to see how fast it is changing.

You can integrate flux, which means finding how much flux has crossed over a certain time. If the field F is constant over time, you can multiply the flux at one instant by your duration. But if F changes with time, then you need to measure at each moment and integrate. Each flux calculation is done at an instant of time, then they are summed together. Again, this is the standard calculus technique.

In our waterfall example, we looked at a single point in time where water had been flowing for a while. If we chose an early point in time, we would have negative flux: water had entered the top, but not yet left the bottom. If we turned off the water, there’d be an instant in time with positive flux: water had stopped entering, but was continuing to leave.

Flux is important for math, electricity and magnetism, and your science life will be better for knowing it. Your social life – not so much.

This was a long article. Take a break. Take a shower. Get outside. See your family. Or, read on about divergence. It’s your call.

Hi Gerard, the dot product is the amount one vector “pushes” in the direction of the other. If they are perpendicular, there is zero push. If they are parallel, there is maximum push. The dot product gives a single number, a scalar.

The cross product is a way to find the “area” spanned by two vectors. In this case, you want them to be perpendicular (to make the largest polygon). If they are parallel, there is no “area” between them. The cross product gives another vector, which is perpendicular to the input vectors.

thank you for your scalar/vector explanation. another puzzle for me has been the bernoulli theorem if it’s valid how does a plane fly when itis inverted? wouldn’t it tend to be forced downward and crash if the higher pressure on the wing is on top?

@Gerard: Thanks for the comment. Unfortunately, I don’t know much about Bernoulli’s theorem, but I’ve read that the angle of attack is as or more important than the exact wing shape (thus flying upside down works with the correct angle of attack).

thanks kalid for the bernoulli answer. how about another mystery — if the purpose of finding solutions to ordinary differential equations is to find a function answer rather than a numerical one, do partial diff eqs find partial function answers?

A minor point: In the Flux Factors section the analogy with a sail, “like a sail facing directly into the wind”, is awkward.
My first reaction was “Huh? That would be zero flux”. And here’s why…

If you’re picturing a bermuda-rigged ship, to say a sail is “directly into the wind”, doesn’t really make sense. If the *ship* were directly into the wind the sail is catching no wind (and that’s what I first pictured).

(If you were picturing a square-rigger instead, the sails would be backed. That’s just weird.)

So for clarity, drop the analogy, or talk about a ship running before the wind – maximum flux in both Bermudas and Square-riggers.

Thanks that really helped.
But I have 2 problems, can you please help me out?

1. You said “would you rather have a handful of $5 or $20 bills “flux” into your bank account? Would you rather have a big or little banana come your way? No need to answer that one.”
we’ll wish to have $5 “flux” into our account as flux means passing thru, so if we had $20 fluxed into our account, we’ll be paying more, right?
So, is it what you meant?
But in case of bananas, you said “come your way” but are you considering banana(s) as profit, if that’s so we’ll want the big one other wise the smaller one.
So what do you mean by this?

Okay, sorry I pressed enter by mistake.
Here’s the second thing I was to say.

2. You said “Flux is a total, and is not “per unit area” or “per unit volume”. Flux is the total force you feel, the total number of bananas you see flying by your surface”
But later you said “your field could represent bananas-per-second, in which case you’d get the bananas-per-second crossing your surface. The units of flux depend on the units of your vector field.”

Now what’s that? You are contradicting yourself and what’s more you are confusing us ( or atleast me ).
Please make it clear. Is there a unit or not?

My book says :
The surface integral (integral sign and then A.dS with bars over them to show that they are vectors)represent ‘flow of flux’ of vector field A over surface S. Say, if A = pV (A and V vectors) where p is density and V is velocity of fluid then surface integral (the integral) represent amount of fluid flowing through given surface in unit time.

You see, it says “flow of flux” now what does that means?
Also it says “in unit time” so does flux indeed have units, or not as you first said (then you contradicted yourself).
Please explain, I m very confused.

1) Yes, in both cases (bananas and dollars) the intent for the analogy is for both types of flux to be “good”. So I mean to say that bigger is better in both cases.

2) Good question — I might need to go back and clarify. Flux is defined as the total impact of the field over the entire surface, and not a “piece by piece” impact.

The unit is Force * Area, so you’d have to multiply out the units for whatever force and area represent in your particular situation. In some cases, the “Force” will represent a velocity (m/s), work done, or an amount of something passing through a single point (bananas per second, through this exact point). I’ll need to clarify but flux represents a “multiplication” of force across a certain area. The units of that multiplication will depend on the problem, but in general flux is the total impact of the force on the entire area (not the impact of the force on one particular point).

@Wendy: You’re more than welcome, really glad it helped! I like Feynman, I hadn’t heard that quote — there are so many things that we know on a surface level but don’t understand. Every day I’m seeing more things which I thought I “knew” :). Thanks for writing.

hats off to you for what you are posting on this site.whatever postings I have read of your’s have helped me in getting the physical meaning of various things.Right now i am learning about electromagnetic waves.would be very grateful to you if you would write about the vector calculus used in analysis of electromagnetic waves.
Thanks!!!!!

Although I like the basic explanation I think you overlooked one thing: what you call flux is in some domains called ‘fluence’. The fluence ‘rate’ is called ‘flux’. Just have a look at the Wikipedia pages. Since I’m working in a radiology environment where X-rays are going through areas, I’m used to think about flux as the amount of X rays going through an area in a certain time.

@Gawin: Great question. The surface is an invisible boundary inside the field we’re considering (i.e., if you have a soccer goal, the entire front plane of the goal is an invisible boundary, and the “flux” might be the quantity of soccer balls passing through).

@Nick B: Great point. There may be some subtleties of flux vs fluent that I’m missing. Flux itself might be a rate (bananas/sec) vs just the amount going through (bananas). It likely depends on the application.

@azhar: The banana is whatever your field represents. Usually we talk about an abstract “field” but I prefer to imagine something passing through a boundary. It could be bananas, water, gravitational pull, etc. (bananas are easier to visualize than gravitational pull).

hi,
I am studying this subject but I’m struggling to find whether there is a negative or a positive flux. because if you change the poles of a magnet, the flux also changes, what is the relation between them ?

@azhar: No problem! Yep, banana is the fruit — I just used it as a silly example of something that can pass through a surface. We usually talk about electric fields, but something more tangible might be simpler to visualize.

@mustafa: Flux can be positive or negative, which represents whether the field is entering or leaving the surface. Usually with magnetic fields, we imagine the North pole “emitting” the field and the South pole absorbing it. So if you flip the poles of a magnet, the direction of the field changes, and it goes from emitting to absorbing (in the region we are measuring the flux).

Thank you so much for writing up this series! I must send my 11 year old nephew here. I’m on a self study path from GCSE to post degree – might take me a while but your site, I think, will shorten how long it takes for me. So, nice one!

First comment of this year .Whoa! terrific work . I had been looking for such type of perspective on science/maths. I have shared it with with friends. It was also interesting to read the comments and how people pointing out the nitty gritties and mistakes and more importantly,your patient replies.
Actually,I was searching for the similar type of perspective on chemistry.THOSE HELL EQUATIONS.:P all rote!

I love your blog and your imagination. Your examples definitely help to integrate the theory and they’re often wonderfully silly which is a much needed break from the regular abstract rigorous mathematical texts.

Most modern textbooks are awful. There’s nothing worse than loving mathematics and going to class and being given all the answers with no ounce of motivation or applicability.

Thanks MR, really appreciate it. Yes, so many textbooks seem to forget what it was like being a student who wanted to genuinely understand and enjoy the material. It’s never too late to make our own versions though :).

Someone posted about the ‘flow of flux’.
Several textbooks use this terminology and it has always driven me nuts. Certainly the flux itself is not flowing. The flux exists and tells us how much flow we have of the ‘whatever-it-is’.

Negative flux means something is entering the region, like pouring water *into* a bucket. Zero flux means nothing is entering or leaving, like trying to pour water into a bucket that is sideways (nothing enters).

From a human perspective, what we call “negative flux” is typically better than zero flux. It depends on the scenario.

I’ve got a glimpse about flux in high school physics (Electric flux and Gauss Law of E. fields without even using regular calculus). I thought a little and understood flux in a way. Tell me if those are correct. (I have yet to study Vector calculus in undergrad)

* Why flux is used in physics?: Fields are creepy. We are not very used to force-at-a-distance. Imagine trying to touch ur friend and getting repelled by some kind of mysterious force, how creepy is that? But the same thing happens in electromagnetism, so we are not very happy about it. On the other hand, you try to touch a working table-fan and you feel it pushing u away at a distance. We don’t freak out, bcz we know that fan is pushing (emitting) the air and the emitted-yet-invisible air is pushing us away.

So, we are trying to use the same analogy in fields. We feel better in assuming that a charge is ‘emitting’ something, and that something pushes another like charge near it. Of course, we know that charge isn’t actually emitting anything, bcz it’s energy is not reducing with time. But we love to think of it that way. We call this imaginary ‘something’ that’s emitted: Flux.

To understand the implications of Gauss law (Flux depends ONLY in the charge inside. Not the charges outside, not the position of the charge inside, not the size of surface…), I imagine the charge as the sun. Sun emits power (energy per unit time) in all directions, just like the charge emits flux. (Although flux is imaginary & power is real). If we draw an imaginary closed surface around the sun, the TOTAL power passing through surface doesn’t depend on whether the sun is in the center of the surface or not. Also, it won’t depend on the size of surface. A surface with smaller radius will have high intensity (power per unit surface area) but will have the same TOTAL power passing through it. Now, If we place the sun outside the surface, all the power that passes inside will pass outside. So we can say, the NET total power through the surface is zero. And that are the implications of the Gauss law.

To understand the concept that [flux = field intensity x area x cos(orientation angle)], I like to think flux as river flow (sun’s energy is radiated spherically, so can’t take it as analogy here) as u said. orientation of the bucket decides how much water is collected.